By the Wedderburn–Malcev theorem for the ordinary case (see for example [1]}), we can write \(A\)  as a direct sum of vector spaces
\(A = B + J = B_1 \oplus \cdots \oplus B_k + J\) 
where \(B\)  is a maximal semisimple subalgebra of \(A\) , \(J= J(A)\)  is the Jacobson radical of \(A\) , and \(B_i\)  are simple algebras, \(i = 1\) , ..., \(k\) . By the Theorems of Wedderburn and Wedderburn–Artin on simple and semisimple algebras (see for instance [2]}), and since \(F\)  is algebraically closed, we have that
\(B = B_1 \oplus \cdots \oplus B_k = M_{n_1}(F) \oplus \cdots \oplus M_{n_k}(F).\) 
Here \(M_{n_i}(F)\)  is the simple algebra of \(n_i \times n_i\)  matrices, \(i = 1\) , ..., \(k\) . Clearly \(B\)  is a trace-subalgebra since \(1_A \in B\) . Moreover, by considering the restriction of the trace \(\mbox{tr}\)  on \(B\)  it is easy to see that there exist \(\alpha _i \in F\)  such that
\(tr(a_1, \ldots , a_k) = \sum _{i = 1}^{k} t_{\alpha _i}(a_i)\) 
where \(a_i \in M_{n_i}(F)\) , \(t_{\alpha _i} = \alpha _i t_1^i\) , and \(t_1^i\)  is the ordinary trace on the matrix algebra \(M_{n_i}(F)\) .
